<h2>Problem 200</h2>
<div style="color:#666;font-size:80%;">28 June 2008</div><br />
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<p>We shall define a sqube to be a number of the form, <var>p</var><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /><var>q</var><img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" />, where <var>p</var> and <var>q</var> are distinct primes.<br />
For example, 200 = 5<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />2<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" /> or 120072949 = 23<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" />61<img src="" style="display:none;" alt="^(" /><sup>3</sup><img src="" style="display:none;" alt=")" />.</p>

<p>The first five squbes are 72, 108, 200, 392, and 500.</p>

<p>Interestingly, 200 is also the first number for which you cannot change any single digit to make a prime; we shall call such numbers, prime-proof. The next prime-proof sqube which contains the contiguous sub-string &quot;200&quot; is 1992008.</p>

<p>Find the 200th prime-proof sqube containing the contiguous sub-string &quot;200&quot;.</p>
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